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%TCIDATA{Created=Friday, September 19, 2003 12:48:26}
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\begin{document}

\title{Ordinal logistic regression; A comparison with software for
multilevel modelling}
\author{}
\maketitle

\paragraph{\protect\bigskip Model description}

\bigskip In this model the response variable $y$ takes on values from the
ordered set $\{y^{(s)},s=1,\ldots ,S-1\}$, where $y^{(1)}<y^{(2)}<\cdots
<y^{(S)}$. For $s=1,\ldots ,S-1$ define $P_{s}=P\left( y\leq y^{(s)}\right) $
and $\kappa _{s}=\log [P_{s}/(1-P_{s})]$. To allow $\kappa _{s}$ to depend
on covariates specific to the $i$th observation ($i=1,\ldots n$) we
introduce a disturbance $\eta _{i}$ of $\kappa _{s}$:

\begin{equation}
P\left( y_{i}\leq y^{(s)}\right) =\frac{\exp (\kappa _{s}-\eta _{i})}{1+\exp
(\kappa _{s}-\eta _{i})},\qquad s=1,\ldots ,S-1.
\end{equation}%
with%
\begin{equation}
\eta _{i}=\mathbf{X}_{i}\mathbf{\beta }+u_{j_{i}},
\end{equation}%
where $\mathbf{X}_{i}$ and $\mathbf{\beta }$ play the sample role as in
Example 1-3, the $u_{j}$ ($j=1,\ldots q$) are independent $N(0,\sigma ^{2})$
variables, and $j_{i}$ is the latent variable class of individual $i$.

The SOCATT data set is used in a software review conducted by Centre for
Multilevel Modelling (http://multilevel.ioe.ac.uk/softrev/index.html), in
which different packages for multilevel modelling were compared. The SOCATT
data consist of responses to a set of dichotomous items on a woman's right
to have an abortion under different circumstances. The outcome variable $y$
is a score constructed from these items ranging from 1 to 7, with a higher
score corresponding to stronger support for abortion. Each of $q=264$
respondents was asked the same set of questions on four occasions (hence $%
n=1056$) in the period $1983-1986$, and $y_{ij}$ denotes the response for
individual $i$ at year $j$. We consider one categorical covariate, religion
(1=Roman catholic, 2=Protestant/Church of England, 3=Other, 4=None),
represented by three dummy variables for the last three categories ($x_{1j}$%
, $x_{2j}$, $x_{3j}$, respectively). A random intercept ordered logit model
was fitted: 
\begin{equation}
\eta _{i}=\beta _{1}\,x_{i1}+\beta _{2}\,x_{i2}+\beta _{3}\,x_{i3}+u_{i},
\label{eta_ordinal}
\end{equation}%
with $u_{i}\sim N(0,\sigma ^{2})$.

\paragraph{Results}

Estimates of hyper-parameters are shown in the following table:

\begin{center}
\begin{tabular}{lllllllllll}
& $\beta _{1}$ & $\beta _{2}$ & $\beta _{3}$ & $\sigma $ & $\kappa
_{1}\qquad $ & $\kappa _{2}$ & $\kappa _{3}$ & $\kappa _{4}$ & $\kappa _{5}$
& $\kappa _{6}$ \\ \hline
ADMB-RE & 1.953 & 0.684 & 2.775 & 2.229 & -4.127 & -2.390 & 0.402 & 1.337 & 
2.225 & 3.265 \\ 
aML & 2.064 & 0.688 & 2.841 & 2.283 & -4.056 & -2.300 & 0.510 & 1.449 & 2.341
& 3.384%
\end{tabular}
\end{center}

The computaton time (ADMB-RE)\ for this model was 30 seconds on a 1,400 MHz
PC running linux, while for the packages participating in the sofware review
the computation times ranged from 5 to 60 seconds.

\end{document}
